1 3 Heat Mass Transfer DOI 10.1007/s00231-016-1844-z ORIGINAL Analysis of polydisperse fuel spray flame Ophir Nave 1,2 · Yaron Lehavi 3 · Suraju Ajadi 4 · Vladimir Gol’dshtein 1 Received: 21 April 2015 / Accepted: 27 May 2016 © Springer-Verlag Berlin Heidelberg 2016 L Liquid evaporation energy (i.e., latent heat of evapo- ration, enthalpy of evaporation) Le Lewis number m Mass N Number of collisions P Probability p Pressure Q Heat of reaction Q i i = 0, 1, 2, 3 the distribution function of the ith moment R Radius of droplet T Temperature u Velocity V Volume v Average relative velocity * Denotes the non dimensional parameter Greek symbols ρ Density η Dimensionless parameter defined in Eq. (2.11) μ Dimensionless parameter defined in Eq. (2.11) α Dimensionless parameter defined in Eq. (2.11) and refer to volumetric droplet content  Thermal conductivity Subscripts coll Collision cell Cell of parcel F Fuel f Flame rel Relative 0 Refers to condition at x = 0 O Oxygen g Gas Abstract In this paper we analyzed the model of polydis- perse fuel spray flame by using the sectional approach to describe the droplet-droplet interaction within the spray. The radii of the droplets are described by a probability den- sity function. Our numerical simulations include a com- parative analysis between three empirical droplet size dis- tributions: the Rosin–Rammler distribution, the log-normal distribution and the Nakiyama–Tanasawa distribution. The log-normal distribution was found to produce a reason- able approximation to both the number and volume size distribution function. In addition our comparative analysis includes the application of the homotopy analysis method which yields convergent solutions for all values of the rel- evant parameters. We compared the above results to experi- mental fuel spray data such as Tetralin, n - Decane, and n - Heptane. List of symbols A Pre-exponential rate factor B Universal gas constant C Vaporization coefficient E Activation energy c p Specific heat of capacity D Diffusion * Ophir Nave naveof@cs.bgu.ac.il 1 Department of Mathematics, Ben-Gurion University of the Negev (BGU), Beersheba, Israel 2 Jerusalem College of Technology (JCT), Jerusalem, Israel 3 Department of Physics, The David Yellin Academic College, Jerusalem, Israel 4 Department of Mathematics, Faculty of Science, O.A.U, Ife, Nigeria